16.3. Resolution of Herrera's problem 425Pn was finite-rank, we have Q(P,;{:1-i) = Q(7-i) (canonically) and the proof is
complete. DWe need a well-known theorem of Larry Brown. (See [53, Theorem
III.6.3] for a proof.)
Theorem 16.2.5. Let 0 __, J __, A __, B __, 0 be a short exact sequence
where both J and B are AF algebras. Then A is also AF. In other words,
extensions of AF algebras are AF.Finally, we present the generalization of Theorem 7.5. 7 that we've been
after.
Theorem 16.2.6. Let A be an exact C -algebra and a: A __, B(7-i) be a
quasidiagonal representation. For each finite set J C A and c: > 0 there
exists a finite-dimensional C-subalgebra B c B(7-i) such that
a(J) cc B.
Proof. Fix a finite set i c A and c: > 0. As before, 7r: B(7-i) __, Q(1-i) is the
canonical quotient map. Let D c Q(1-i) be a finite-dimensional subalgebra
such that
7r o a(J) cc D.
Let C c B(7-i) be the canonical extension of IK(7-i) by D (i.e., the pullback
of D c Q(7-i) in B(7-i)). Since C is AF, it suffices to show that
a(J) cc 0,
which is easy, since IK(7-i) C C. D
16.3. Resolution of Herrero's problem
To resolve Herrera's problem, we must consider another natural class of
operators.
Definition 16.3.1. An operator S E B(7-i) is called banded if there exists
an orthonormal basis {Vi} of 1-i such that the m~trix of S with respect to
{vi} is banded (meaning only a finite number of diagonals are nonzero -
that is, there exists N EN such that (Svi, VjJ = 0 whenever Ji - jJ > N).
We let Band(7-i) denote the set of banded operators on 7-i.^1
In order to apply Theorem 16.2.6, we will need a simple observation.
Lemma 16.3.2. If TE Band(H), then C*(T) is exact.11n our previous terminology, these are finite propagation operators, or a·perators'supported
in tubes.